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《山东大学学报(理学版)》 ›› 2019, Vol. 54 ›› Issue (7): 89-99.doi: 10.6040/j.issn.1671-9352.0.2018.588

• • 上一篇    

一类具有随机效应的SIRI传染病模型的定性分析

高建忠,张太雷*   

  1. 长安大学理学院, 陕西 西安 710064
  • 发布日期:2019-06-27
  • 作者简介:高建忠(1991— ), 男, 硕士研究生, 研究方向为常微分方程及其应用. E-mail: gaojianzhong2017@126.com*通信作者简介:张太雷(1980— ), 男, 博士,教授, 研究方向为常微分方程及其应用. E-mail: t.l.zhang@126.com
  • 基金资助:
    国家自然科学基金资助项目(11701041);陕西省自然科学基础研究计划项目(2018JM1011;2017JQ1014)

Qualitative analysis of an SIRI epidemic model with stochastic effects

GAO Jian-zhong, ZHANG Tai-lei*   

  1. School of Science, Changan University, Xian 710064, Shaanxi, China
  • Published:2019-06-27

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摘要： 研究了具有随机效应的SIRI双线性传染病模型。利用停时理论及Lyapunov分析方法, 证明了随机模型正解的全局存在唯一性和有界性, 讨论了随机模型的解在相应确定性模型的无病平衡点和地方病平衡点附近的振荡行为, 得到了随机模型的解的平均持续和疾病灭绝的充分条件。最后, 数值模拟验证了理论结果的正确性。

关键词: 随机SIRI模型, 振荡行为, 平均持续, 疾病灭绝

Abstract: An SIRI bilinear epidemic model with stochastic effects is studied. The global existence, uniqueness and boundedness of its positive solution are proved by using stopping time theory and Lyapunov analysis method. It is also shown that the solution of the stochastic model oscillates around the corresponding deterministic disease-free equilibrium and endemic equilibrium points, and the sufficient conditions for persistence in mean of the solution of the stochastic model and disease extinction are obtained. Finally, numerical simulations are carried out to prove the validity of theoretical results.

Key words: stochastic SIRI model, oscillating behavior, persistence in mean, disease extinction

中图分类号: 

  • O175.1
[1] TUDOR D. A deterministic model for herpes infections in human and animal populations[J]. Siam Review, 1990, 32(1): 136-139.
[2] GUO Peng, YANG Zhichun. The stability of SIRI model with nonliner incidence rate[J]. Journal of Chongqing Normal University: Natural Science, 2011, 28(4): 35-39.
[3] BLOWER S. Modelling the genital herpes epidemic[J]. Herpes, 2004, 11(3): 138-146.
[4] BELOTTO A, LEANES L F, SCHNEIDER M C, et al. Overview of rabies in the Americas[J]. Virus Research, 2005, 111(1): 1-12.
[5] NAZ R, MAHOMED K S, NAEEM I. First integrals and exact solutions of the SIRI and tuberculosis models[J]. Mathematical Methods in the Applied Sciences, 2016, 39(15): 4654-4666.
[6] 雷桥, 杨志春. 具有随机效应的SIRI传染病模型的定性分析[J]. 重庆师范大学学报(自然科学版), 2016, 33(1): 82-85.LEI Qiao, YANG Zhichun. The qualitative analysis of SIRI epidemic model with stochastic effects[J]. Journal of Chongqing Normal University(Natural Science), 2016, 33(1): 82-85.
[7] CAI Yongli, KANG Yun, WANG Weiming. A stochastic SIRS epidemic model with nonlinear incidence rate[J]. Applied Mathematics and Computation, 2017, 305: 221-240.
[8] ZHAO Yanan, JIANG Daqing. The threshold of a stochastic SIRS epidemic model with saturated incidence[J]. Applied Mathematics Letters, 2014, 34(1): 90-93.
[9] LIU Qun, JIANG Daqing, SHI Ningzhong, et al. The threshold of a stochastic SIS epidemic model with imperfect vaccination[J]. Mathematics and Computers in Simulation, 2017, 144: 78-90.
[10] YANG Qingshan, JIANG Daqing, SHI Ningzhong, et al. The ergodicity and extinction of stochastically perturbed SIR and SEIR epidemic models with saturated incidence[J]. Journal of Mathematical Analysis and Applications, 2012, 388(1): 248-271.
[11] 刘杰, 胡志兴, 廖福成. 随机模型SIRS的定性分析[J]. 宁夏大学学报(自然科学版), 2016, 37(1): 1-6. LIU Jie, HU Zhixing, LIAO Fucheng. Qualitative analysis of stochastic SIRS model[J]. Journal of Ningxia University(Natural Science Edition), 2016, 37(1): 1-6.
[12] 张艳宏, 许超群, 原三领. 一类接触率受到噪声干扰的随机SIS流行病模型研究[J]. 上海理工大学学报, 2015, 37(6): 512-516. ZHANG Yanhong, XU Chaoqun, YUAN Sanling. Stochastic SIS epidemic model with contract rate influenced by noise[J]. Journal of University of Shanghai for Science and Technology, 2015, 37(6): 512-516.
[13] LIU Qun, JIANG Daqing, HAYAT T, et al. Stationary distribution and extinction of a stochastic SIRI epidemic model with relapse[J]. Stochastic Analysis and Applications, 2018, 36(1): 138-151.
[14] JI Chunyan, JIANG Daqing, SHI Ningzhong. The behavior of an SIR epidemic model with stochastic perturbation[J]. Stochastic Analysis and Applications, 2014, 30(5): 755-773.
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